How do I get Particle Boids to fight each other?
You need to define at least 2 particle systems with a relation enemy to each other, you can define attributes like strength (this will statistically decide who wins [particle disappears when defeated]). 

You could add Follow Leader to the boids brain and animate the leader, this would make the particles follow the leader (creates nice formations). There're so many options you need to play around and develop an idea of how it should like.

 

In the physics settings make sure you checked Allow Flight or restrict them to land.



I stopped playing with this because I wasn't able to create explosions when a particle dies.

Example: